Phase retrieval and diffractive imaging based on Babinet's principle and complementary random sampling

Zhen-Jia Cheng; Ben-Yi Wang; Yi-Yan Xie; Yu-Jie Lu; Qing-Yang Yue; Cheng-Shan Guo

doi:10.1364/OE.23.028874

1. Introduction

Coherent diffractive imaging (CDI) [1] is a lensless imaging technique for 2D or 3D reconstruction of an object from its diffraction intensities. It is very attractive for many applications because it can retrieve the phase information of the object and can avoid using high-quality imaging lenses. A key issue in the CDI is to retrieve the phase information of the object wavefront from its diffraction intensity patterns. Many approaches have been explored and developed for this purpose. The most commonly used methods are perhaps those based on the holographic interferometry, such as in-line or off-axis holography [2–10], in which a suitable reference component or phase-shifting algorithm must be required. Another approach is the use of iterative methods [11–30], where no separate reference beam is needed but the phase information of the tested wavefront is iteratively retrieved from the diffraction intensities. The conventional iterative algorithms usually require a large number of iterations and may not lead to a unique solution due to intrinsic non-uniqueness of the standard phasing problem and the competition among the true and the ambiguous solutions. Sampling the diffraction patterns of the objects more finely than the Nyquist frequency [14], using more than one of the diffraction patterns [15–22] and multiple structured illuminations [23,24], and modulating the test wavefront with a phase plate [25–30] can improve the phase retrieving process and make the retrieved phase uniquely. Despite tremendous progresses, however, many questions, fundamental as well as algorithmic, remain to be solved. For example, in order to constrain the field in the iterative phase retrieval process, the object must be much smaller than the image sensor used.

Recently, Horisaki et al. [31,32] proposed a CDI method based on a binary amplitude mask with randomly sampled pinholes. They verified by numerical simulations that the problems of non-uniqueness and non-convergence in the phase retrieval can be solved by introducing a binary amplitude random sampling (RS) mask between the object and the image sensor. For simplicity, this method will be abbreviated as RS-CDI method in the following texts. This RS-CDI method effectively alleviates the limitation regarding the object’s size in conventional CDIs. However, only a fraction of the object wavefront can be retrieved in the RS-CDI method because of the existence of the binary RS mask between the object and the recording plane. Although imaging the object from the sieved field could be realized based on compressive sensing algorithm [33], high redundancy of the tested field must be required.

In this paper we propose a method for phase retrieval based on Babinet's principle and complementary random sampling (CRS) [34,35]. In this method, a group of specially designed CRS masks are, respectively, inserted between the object and the image sensor; the whole (not sieved) phase distribution of the field at the sampling plane can be retrieved from the recorded intensity patterns based on Babinet's principle; and the image of the object can be realized just through a single inverse digital diffraction from the sampling plane to the object plane.

2. Principle and simulation

Figure 1 shows a general schematic for phase retrieval and diffractive imaging based on random sampling. Here an object is illuminated by a spatially coherent plane wave with a wavelength of λ. The object plane is defined as the plane immediately behind the object. The complex amplitude of the object wave on this object plane can be generally expressed as $o (x_{o}, y_{o})$ , which is the complex function to be imaged. In the RS-CDI method proposed by Horisaki et al. [31], a RS mask with some randomly distributed pinholes is inserted between the object plane and the recording plane as shown in Fig. 1. The diffraction intensity pattern of the object wave sieved by the RS mask is recorded by a CCD sensor at the recording plane. Supposing the distance z between the sampling plane and the recording plane satisfies the Fresnel approximation, the intensity distribution on the recording plane at the distance of z downstream from the sampling plane can be written as

I (u, v) = {| F r_{z} {S (x, y) o (x, y)} |}^{2},

where

F r_{z} {}

indicates the Fresnel integral with the diffraction distance of z, which is also named as the Fresnel transform,

o (x, y)

is the complex amplitude of the object wave propagating to the sampling plane from the object

o (x_{o}, y_{o})

; and

S (x, y)

denotes the transmittance function of the RS mask, which is a binary amplitude transmittance function.

Fig. 1 Schematic diagram for lensless diffractive imaging based on random sampling.

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Horisaki et al. [31] have demonstrated by computer simulations that the sieved field $S (x, y) o (x, y)$ in Eq. (1) can be high efficiently retrieved from the intensity $I (u, v)$ using a standard iterative algorithm such as the input-output iteration algorithm [13], that is to say, the sieved field can be approximately expressed as

S (x, y) o (x, y) \approx I t e r {I (u, v), S (x, y)}

in which,

I t e r {}

denotes the iterative operation adopted between the sampling plane and the recoding plane for retrieving the sieved field from the known intensity

I (u, v)

at the recording plane by using

S (x, y)

as the support constraint at the sampling plane. However, in the RS-CDI method, a considerable amount of wavefront information of the tested field was lost because of the insertion of the binary RS mask. For example, only a quarter of wavefront information could be retrieved when the opening ratio of the RS mask is designed to its optimal value of 0.25. Although the quality of the image from the sparsely sampled field could be greatly improved by using compressive sensing algorithm [33], a further iterative operation must be carried out between the object plane and the sampling plane, and so some further limitations to the object should be enforced.

Here we suggest another more effective solution for eliminating the influence of the binary sampling mask on the retrieved image. Because this method is mainly based on the Babinet's principle and the concept of the CRS, hereafter we will abbreviate this method as CRS-CDI method for simplicity.

At first we introduce the concept of the CRS. Suppose there are a group of RS masks with the mask number of N. Their transmittances $S_{n} (p, q)$ are defined as

S_{n} (p, q) = {\begin{matrix} 1, for R (p, q) = n \\ 0, for R (p, q) \neq n \end{matrix}, (n = 1, 2, \dots, N),

in which, (p,q) are two dimensional (2-D) discrete coordinates, and

R (p, q)

is a 2-D array of integers randomly distributed from the integer 1 to the integer N. Obviously, such a group of masks are complementary, which means that they satisfy

\sum_{n = 1}^{N} S_{n} (p, q) = 1.

So we name this group of RS masks satisfying Eqs. (3) and (4) as CRS masks.

Placing one of the CRS masks on the sampling plane of Fig. 1, the corresponding diffraction intensity at the recording plane can be recorded by an image sensor. The recorded intensity corresponding to the n-th mask can be expressed as

I_{n} (u, v) = {| F r_{z} {S_{n} (p, q) o (p, q)} |}^{2}, n = 1, 2, \dots, N .

Then the corresponding complex amplitude sieved by each of the CRS masks can be retrieved respectively by using the iterative algorithm described by Horisaki et al. in [31], which can also be approximately expressed as

{\tilde{o}}_{n} (p, q) = S_{n} (p, q) o (p, q) \exp (i δ_{n}) \approx I t e r {I_{n} (u, v), S_{n} (p, q)}, n = 1, 2, \dots, N,

where δ_n is the phase bias introduced by the initial phase conditions of the iterative operations. Our further simulation and experimental results reveals that, if the iterative operations for all the intensities

I_{n} (u, v)

are kept in a same initial phase condition, the phase bias can remain unchanged, namely

δ_{n} \approx δ_{0}

(δ₀ is a constant). Then the original object wavefront

o (p, q)

at the sampling plane can be found out according to the Babinet's principle [34] and the properties of the CRS masks given in Eq. (4); that is,

\sum_{n = 1}^{N} I t e r {I_{n} (u, v), S (p, q)} \approx o (p, q) \sum_{n = 1}^{N} S_{n} (p, q) \exp (i δ_{n}) = o (p, q) \exp (i δ_{0}) .

From Eq. (7) we can see that, because of the introduction of the CRS masks, the influence of the random sampling on the tested field is successfully eliminated. Using such retrieved complex amplitude (including the phase and the amplitude), the image of the object can be reconstructed just by a single inverse digital diffraction operation from the sampling plane to the object plane, even if the object is complex-valued or three-dimensional.

The CRS-CDI method described above was firstly demonstrated by computer simulations. In our simulations, the sampling numbers of the object plane, the sampling plane and the recording plane are respectively taken as $512 \times 512$ , and the sampling interval is set to be 18 μm. The test object is a phase object with the phase distribution as shown in Fig. 2(a) (only the central $300 \times 300$ pixels are shown). The object is assumed to be illuminated by a plane beam with a wavelength of $λ = 632.8 n m$ . Figures 2(b)-2(e) are a group of CRS masks adopted in the simulations, which were designed according to Eq. (3) when the opening ratio (the ratio of the white pixel number to the total pixel number) of each mask is set to be 0.25, corresponding to the mask number N = 4). These CRS masks are placed on the sampling plane of Fig. 1 respectively, and the corresponding diffraction intensities at the recording plane are recorded as the known information. Figure 2(f) is an example of the recorded intensity distributions of the object wave propagating to the recording plane through a CRS mask when the distances from the object plane to the sampling and from the sampling plane to the recording plane are set to be, respectively, 300mm and 200mm.

Fig. 2 Simulation results. (a) Phase distribution of the pure phase object to be measured. (b)-(e) Prepared a group of CRS masks with N = 4. (f) An example of the intensity distributions obtained in the recording plane. (g) and (h) Extracted amplitude and phase distributions of the object wave at the sampling plane based on our CRS-CDI method. (i) Phase distribution of the reconstructed image through an inverse diffraction of the extracted complex amplitude shown in (g) and (h). (j) An example of the reconstructed image through an inverse digital diffraction of the sieved field retrieved by the RS-CDI [25] using the intensity shown in (f). (k) An imaging example reconstructed when the initial phase of the iterative operations is randomly changed with different known intensity $I_{n}$ .

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Figures 2(g) and 2(h) show the extracted amplitude and phase distributions of the object wave based on our CRS-CDI method with the following procedures: (a) Set the initial complex amplitude at the recording plane as the known amplitude $\sqrt{I_{n}}$ with an initial guess phase distribution (it should be same for each $\sqrt{I_{n}}$ in our method). (b) Back-propagate the complex field from the recording plane to the sampling plane. (c) Multiply the back-propagated fields by the corresponding CRS function $S_{n}$ to form the guess retrieved field ${\tilde{o}}_{n}$ at the sampling plane, and then forward-propagate ${\tilde{o}}_{n}$ again to the recording plane. (d) Replace the amplitude of the forward-propagated field on the recording again by $\sqrt{I_{n}}$ . (e) Iteratively repeat steps (b) to (d) until ${\tilde{o}}_{n}$ converges. (f) Add all ${\tilde{o}}_{n}$ to form the final retrieved amplitude and phase distributions as shown in Figs. 2(g) and 2(h). The iteration times run for the result are taken as 30. Finally, the image of the object can be reconstructed through further back-propagating the retrieved field shown in Figs. 2(g) and 2(h) from the sampling plane to the object plane. The final reconstructed image is shown in Fig. 2(i).

As comparison, Fig. 2(j) gives an example of the reconstructed image through an inverse digital diffraction of the sieved field retrieved by the RS-CDI [28] method using one intensity pattern as shown in Fig. 2(f). Here the iteration times are also taken as 30. We can see that there exists a severe random noise in the reconstructed image; the main reason is that a considerable amount of wavefront information of the object was lost because of the insertion of a binary sampling mask. For a sparsely sampled field with high redundancy, this kind of noise could be eliminated by using a compressive sensing algorithm [31,33].

From the results shown in Figs. 2(i) and 2(j) it can be seen that the sampling noise is successfully eliminated and thus the final imaging quality is highly improved. It should be indicated that, because the amplitude outside the central circular part of the object is equal to zero in the simulations, the retrieved phase in these areas is meaningless.

For illustrating the importance of the initial phase setting at the beginning of the iterative operations, Fig. 2(k) shows an imaging example reconstructed when the initial guess phase of each iterative operation is randomly changed with different known intensity $I_{n}$ . In this case, the phase bias δ_n (given in Eq [6].) introduced by the iterative operation will shift also at random, which results in the severe random noise in the reconstructed image as shown in Fig. 2(k). Our simulation results reveal that, for different intensity $I_{n}$ recorded in our CRS-CDI, the phase bias δ_n given in Eq. (6) can remain the same if the initial guess phase distribution is set to the same one in the beginning of the iterative operations.

In order to quantitatively evaluate the reconstructed phase, Fig. 3 further shows the phase profile of the reconstructed image along the red line in Fig. 2(i). As comparison, the corresponding phase profile of the original object shown in Fig. 2(a) is also given. We can see that the reconstructed phases based on our method are consistent with the phases of the original object except for those at the sharp edges because of the diffraction loss due to the limited aperture in the simulations.

Fig. 3 Phase profiles of the original object (black line) and the reconstructed image (red dashed line) along the red lines shown in Figs. 2(a) and 2(i).

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3. Experiments and discussions

For further demonstrating the practical feasibility of the CRS-CDI method in experiments, we established a simple lensless CDI experimental setup based on a commercial transmission-type spatial light modulator (SLM) and made the experiments to test the method. In principle, the CRS-CDI method is applicable to any wave field, such as infrared, visible light, X-rays or electron beams. In our experiment, a He-Ne laser with the wavelength of $λ = 632.8 n m$ is used as the wave source. The object is a transmittance USAF resolution target with group number of 2 to 7. The distance from the object to the sampling plane is set to be about 185mm. The SLM (Sony LCX029) with the pixel number of $1024 \times 768$ and pixel size of $18 μ m \times 18 μ m$ was placed at the sampling plane as shown in Fig. 1, which is set to work in intensity modulating mode (with a contrast ratio of about 300). For generating the required CRS masks in experiments, we respectively display the prepared CRS pictures as shown in Figs. 2(b)-2(e) on the SLM. The diffraction intensities on the recording plane was captured by a commercial 8-bit CCD camera (GYD-SG1300, CAMYU CO. LTD) with pixel size of 6.7μm and pixel number of $1030 \times 1300$ .

Figures 4(a)-4(d) show the intensity distributions recorded by the CCD camera when the CRS masks (with the number of N = 4) as shown in Figs. 2(b)-2(e) were, respectively, displayed on the SLM. To facilitate precisely locating the RS mask in the subsequent iterative processing, the distance between the sampling plane and the recording plane is set to be about 201mm, which can be easily calibrated according to the Talbot effect through displaying on the SLM an orthogonal grid picture with a period of 14 pixels in experiments. Figures 4(e) and 4(f) show an example of the extracted amplitude and phase distribution at the sampling plane retrieved based on our CRS-CDI method with 30 iterations. Figures 4(g) and 4(h) further give the amplitude and phase distributions of the image at the object plane, which are acquired simply through an inverse digital diffraction of the extracted complex amplitude from the sampling plane back to the object plane. This reconstructed image, although there still exists some noise, demonstrated the feasibility of our CRS-CDI method, in view of the existing experimental condition.

Fig. 4 Experimental results based on the CRS-CDI method. (a)-(d) are the recorded diffraction intensities when the CRS masks with the number of N = 4 are respectively placed on the sampling; (e) and (f) are the retrieved amplitude and phase at the sampling plane using CRS-CDI algorithm; (g) and (h) are the amplitude and phase of the final reconstructed image.

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According to the Abbe theory [36], the theoretical spatial resolution of a diffractive imaging system can be estimated by $Γ = 1.22 λ / N A$ , where NA is the effective numerical aperture of the system. In our experiments, because the effective diameter of the CRS masks is taken as 7.2 mm, the distance between the object to the sampling plane is equal to 185mm and the wavelength of the source is 632.8 nm, the theoretical spatial resolution of the system should be about 20 μm. From our reconstructed image shown in Fig. 4(g), it can be seen that the element 2 (corresponding to a line width of 27.86 μm) of group number 4 in the USAF target can be resolved. We could expect higher imaging resolution and quality if a CCD camera with lower noise and higher dynamic range such as a cooled 12-bit CCD and a SLM with higher contrast ratio and smaller pixel pitch are adopted. In addition, the pixel opening shape and size of the SLM are ignored in the iterative operations, which could be also brought some error to the retrieved results.

Because the CRS-CDI method improves the imaging quality at the expense of increasing the recording times, the next thing to be done after the demonstration of its feasibility is to determine the optimal recording times–that is to say, the optimal number N in a group of the CRS masks. For this purpose, we calculated the reconstruction fidelities of the CRS-CDI with different mask number N characterized by the peak signal-to-noise ratio (PSNR) of the corresponding amplitude of the reconstructed image, which has been shown in Fig. 5. As a comparison, the PSNR of the RS-CDI with a single RS mask is also given in Fig. 5. From Fig. 5 it can be seen that larger number N does not bring higher reconstruction fidelity as it is imagined. An optimal number N of the CRS masks does exist, that is to say, a group of CRS masks with the number of N = 4 should be the most optimal option in practical applications of the CRS-CDI method.

Fig. 5 The PSNR of the CRS-CDI versus the number N of the CRS masks. As a comparison, the PSNR of the RS-CDI with a RS mask is also given (red point).

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4. Conclusion

In conclusion, we have experimentally demonstrated the feasibility of the CRS-CDI method based on Babinet's principle and complementary random sampling. In this method, a group of CRS masks are inserted, respectively, between the object and the recording planes; the whole complex amplitude (not sieved) at the sampling plane can be accurately extracted by the corresponding diffraction intensities with the CRS masks and the object can be digitally imaged through a single inverse diffraction from the sampling plane to the object plane. Our experimental results reveal that inserting the CRS masks between the object and the recording sensor can greatly improve the accuracy and efficiency of the phase retrieval; at the same time, the influence of the random sampling on the sieved field can be well eliminated. Because the iterative process is only carried out between the sampling plane and the recording plane, this method could be suitable for more general complex-valued object without proper constraint. As an on-axis geometry using only binary amplitude sampling masks, this method may provide another potential approach for diffractive imaging or quantitative phase imaging [37–39] in various wavelength regions.

Acknowledgments

The work is supported in part by National Natural Science Foundation of China (NSFC) under Grant No. 11474186 as well as the Research Foundation for the Doctoral Program of Higher Education of China under Grant No. 20113704110002.

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Phase retrieval and diffractive imaging based on Babinet's principle and complementary random sampling

Abstract

1. Introduction

2. Principle and simulation

3. Experiments and discussions

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (5)

Equations (7)

Optics Express